from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7360, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([8,3,12,0]))
pari: [g,chi] = znchar(Mod(323,7360))
Basic properties
Modulus: | \(7360\) | |
Conductor: | \(320\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{320}(3,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7360.ct
\(\chi_{7360}(323,\cdot)\) \(\chi_{7360}(507,\cdot)\) \(\chi_{7360}(2163,\cdot)\) \(\chi_{7360}(2347,\cdot)\) \(\chi_{7360}(4003,\cdot)\) \(\chi_{7360}(4187,\cdot)\) \(\chi_{7360}(5843,\cdot)\) \(\chi_{7360}(6027,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
Field of values: | \(\Q(\zeta_{16})\) |
Fixed field: | 16.16.147573952589676412928000000000000.2 |
Values on generators
\((1151,5061,4417,6721)\) → \((-1,e\left(\frac{3}{16}\right),-i,1)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
\( \chi_{ 7360 }(323, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(1\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) |
sage: chi.jacobi_sum(n)